T-duality rules between D-branes and E-branes

I am trying to understand the time-like T-duality rules in Type II effective actions, and how they relate D-branes to E-branes.
In the case of the metric and dilaton, the Buscher rules are unchanged for this particular case so all goes well. The B-field is set to $0$, however I have a problem for the gauge field coming from the Ramond sector.

In the case of space-like reductions, the rules were given in this paper https://arxiv.org/abs/hep-th/9504081 including the Ramond sector. In his papers about exotic string theories, Hull gives the rules in the case of time-like reductions, and in particular for the gauge field equation (3.10) in https://arxiv.org/abs/hep-th/9807127.

As you can see, the rule says that going from the gauge field of a D-brane to an E-brane, I get a minus sign. However in a follow up paper still by Hull https://arxiv.org/abs/hep-th/9808069, the gauge field for the E-brane does not pick up a minus sign: see (3.3) going to (3.8) (the constant $k$ can be set to $0$).
Moreover, in an explicit example of the D2-brane the same happens:(9.2) becomes (9.5) and no sign change.

At first, I thought that since these actions only contain kinetic terms quadratic in the gauge fields the sign might not matter. However those are bosonic truncations, as soon as fermionic terms are taken into account the overall sign of the gauge field will matter.

Any attempt to help is greatly appreciated :)!